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      "source": [
        "<table class=\"tfo-notebook-buttons\" align=\"left\">\n",
        "  <td>\n",
        "    <a target=\"_blank\" href=\"https://colab.research.google.com/github/google/meridian/blob/main/demo/ROI_mROI_Response_Curves.ipynb\"><img src=\"https://www.tensorflow.org/images/colab_logo_32px.png\" />Run in Google Colab</a>\n",
        "  </td>\n",
        "  <td>\n",
        "    <a target=\"_blank\" href=\"https://github.com/google/meridian/blob/main/demo/ROI_mROI_Response_Curves.ipynb\"><img src=\"https://www.tensorflow.org/images/GitHub-Mark-32px.png\" />View source on GitHub</a>\n",
        "  </td>\n",
        "</table>"
      ]
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    {
      "metadata": {
        "id": "Id_gDKaHxQ-i"
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      "cell_type": "markdown",
      "source": [
        "# **ROI, mROI & Response Curves in Meridian**\n",
        "\n",
        "This notebook showcases how to extract and interpret core business metrics from a trained Meridian model, including **Return on Investment (ROI)**, **marginal ROI (mROI)**, and **Response Curves**.\n",
        "\n",
        "For a detailed overview on each metric\\, please visit this [page](https://developers.google.com/meridian/docs/basics/roi-mroi-response-curves).\n",
        "\n",
        "This notebook utilizes sample data. As a result, the numbers and results obtained might not accurately reflect what you encounter when working with a real dataset.\n",
        "\n",
        "\n",
        "<ol start=\"0\">\n",
        "  <li><a href=\"#install\">Preliminary Setup</a></li>\n",
        "  <li><a href=\"#point-estimates\">Point Estimates</a></li>\n",
        "  <li><a href=\"#credible-intervals\">Credible Intervals</a></li>\n",
        "  <li><a href=\"#roi-mroi-distributions\">ROI, mROI distributions</a></li>\n",
        "  <li><a href=\"#response-curves\">Response Curves</a></li>\n",
        "</ol>\n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "gLUII0gnxVJJ"
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      "cell_type": "markdown",
      "source": [
        "<a name=\"install\"></a>\n",
        "## Preliminary setup: Install library, import data, train model\n",
        "\n",
        "This part of the notebook is preliminary setup to install the Meridian Library and train the model. The core part of the notebook involves understanding the output from the model.\n",
        "\n",
        "1. Make sure you are using one of the available GPU Colab runtimes which is **required** to run Meridian. You can change your notebook's runtime in `Runtime > Change runtime type` in the menu. All users can use the T4 GPU runtime which is sufficient to run the demo colab, free of charge. Users who have purchased one of Colab's paid plans have access to premium GPUs (such as V100, A100 or L4 Nvidia GPU).\n",
        "\n",
        "2. Install the latest version of Meridian, and verify that GPU is available."
      ]
    },
    {
      "metadata": {
        "id": "be7m3FhTD4ne"
      },
      "cell_type": "markdown",
      "source": [
        "### Install library"
      ]
    },
    {
      "metadata": {
        "id": "v7dOWPtZxUNm"
      },
      "cell_type": "code",
      "source": [
        "# Install meridian: from PyPI @ latest release\n",
        "!pip install --upgrade google-meridian[colab,and-cuda]\n",
        "\n",
        "from meridian.analysis import analyzer\n",
        "from meridian.analysis import visualizer\n",
        "from meridian.data import data_frame_input_data_builder\n",
        "from meridian.model import model\n",
        "import numpy as np\n",
        "import pandas as pd\n",
        "# check if GPU is available\n",
        "from psutil import virtual_memory\n",
        "import tensorflow as tf\n",
        "\n",
        "ram_gb = virtual_memory().total / 1e9\n",
        "print('Your runtime has {:.1f} gigabytes of available RAM\\n'.format(ram_gb))\n",
        "print(\n",
        "    'Num GPUs Available: ',\n",
        "    len(tf.config.experimental.list_physical_devices('GPU')),\n",
        ")\n",
        "print(\n",
        "    'Num CPUs Available: ',\n",
        "    len(tf.config.experimental.list_physical_devices('CPU')),\n",
        ")"
      ],
      "outputs": [],
      "execution_count": null
    },
    {
      "metadata": {
        "id": "B-XxYudYD7Om"
      },
      "cell_type": "markdown",
      "source": [
        "### Load Data"
      ]
    },
    {
      "metadata": {
        "id": "HGx2ULKQxXVG"
      },
      "cell_type": "code",
      "source": [
        "df = pd.read_csv(\n",
        "    \"https://raw.githubusercontent.com/google/meridian/refs/heads/main/meridian/data/simulated_data/csv/geo_all_channels.csv\"\n",
        ")\n",
        "\n",
        "builder = data_frame_input_data_builder.DataFrameInputDataBuilder(\n",
        "    kpi_type='non_revenue',\n",
        "    default_kpi_column='conversions',\n",
        "    default_revenue_per_kpi_column='revenue_per_conversion',\n",
        ")\n",
        "\n",
        "builder = (\n",
        "    builder.with_kpi(df)\n",
        "    .with_revenue_per_kpi(df)\n",
        "    .with_population(df)\n",
        "    .with_controls(\n",
        "        df, control_cols=[\"sentiment_score_control\", \"competitor_sales_control\"]\n",
        "    )\n",
        ")\n",
        "\n",
        "channels = [\"Channel0\", \"Channel1\", \"Channel2\", \"Channel3\", \"Channel4\"]\n",
        "builder = builder.with_media(\n",
        "    df,\n",
        "    media_cols=[f\"{channel}_impression\" for channel in channels],\n",
        "    media_spend_cols=[f\"{channel}_spend\" for channel in channels],\n",
        "    media_channels=channels,\n",
        ")\n",
        "\n",
        "data = builder.build()"
      ],
      "outputs": [],
      "execution_count": null
    },
    {
      "metadata": {
        "id": "ZPZMNZmdxcnW"
      },
      "cell_type": "markdown",
      "source": [
        "<a name=\"configure-model\"></a>\n",
        "### Run the Model\n",
        "\n",
        "We will initialize the `Meridian` class and use the `sample_posterior()` method to obtain samples from the posterior distribution."
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Initialize and run the model\n",
        "mmm = model.Meridian(input_data=data)\n",
        "mmm.sample_posterior(\n",
        "    n_chains=10, n_adapt=2000, n_burnin=500, n_keep=1000, seed=0\n",
        ")"
      ],
      "metadata": {
        "id": "P2jprh1KrZ-F"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "metadata": {
        "id": "D5z3FiQ3-7SO"
      },
      "cell_type": "markdown",
      "source": [
        "\n",
        "<a name=\"roi-mroi-response-curve\"></a>\n",
        "\n",
        "## Step 1: Analyze ROI, mROI & Respose Curves\n",
        "\n",
        "Now that the model has been trained, we can use the `Analyzer` class to extract key business metrics. This class is the primary tool for translating the statistical model output into actionable marketing insights."
      ]
    },
    {
      "metadata": {
        "id": "8Q8p9UBx-5pF"
      },
      "cell_type": "code",
      "source": [
        "# Create an instance of the Analyzer class\n",
        "analyzer_instance = analyzer.Analyzer(mmm)"
      ],
      "outputs": [],
      "execution_count": null
    },
    {
      "metadata": {
        "id": "zLKqE90vCPoo"
      },
      "cell_type": "markdown",
      "source": [
        "<a name=\"point-estimates\"></a>\n",
        "### Getting Point Estimates for Quick Analysis\n",
        "\n",
        "We can calculate the **mean** of the posterior distribution to get a point estimate for each channel's ROI and mROI."
      ]
    },
    {
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "3tYuJSfmCOVW",
        "outputId": "447b8117-3105-4fa3-e7be-7c3712e19ff8"
      },
      "cell_type": "code",
      "source": [
        "analyzer_instance.summary_metrics().roi.sel({\"distribution\": \"posterior\", \"metric\": \"mean\"}).to_dataframe()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "             metric distribution       roi\n",
              "channel                                   \n",
              "Channel0       mean    posterior  1.586766\n",
              "Channel1       mean    posterior  0.936712\n",
              "Channel2       mean    posterior  1.418973\n",
              "Channel3       mean    posterior  0.876884\n",
              "Channel4       mean    posterior  1.620968\n",
              "All Channels   mean    posterior  1.208054"
            ],
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        "analyzer_instance.summary_metrics().mroi.sel({\"distribution\": \"posterior\", \"metric\": \"mean\"}).to_dataframe()"
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          "base_uri": "https://localhost:8080/",
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        "id": "PJ7m1F7wzJul",
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      "execution_count": null,
      "outputs": [
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            "text/plain": [
              "             metric distribution      mroi\n",
              "channel                                   \n",
              "Channel0       mean    posterior  0.789475\n",
              "Channel1       mean    posterior  0.470342\n",
              "Channel2       mean    posterior  0.791703\n",
              "Channel3       mean    posterior  0.411681\n",
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          "metadata": {},
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    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "565cecd8"
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      "source": [
        "### Understanding Uncertainty: Credible Intervals\n",
        "\n",
        "A key benefit of Bayesian modeling is measuring uncertainty with credible intervals. A credible interval has a direct, intuitive interpretation. Unlike a frequentist confidence interval, Bayesian models make probabilistic statements about the parameters themselves, rather than the data. For example, a 95% credible interval of [\\$0.43, \\$3.15] for TV ROI means there is a 95% probability that the true ROI for TV is between \\$0.43 and \\$3.15. Such direct probability statements are useful for communicating and decision-making."
      ]
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    {
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
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        "id": "cQXa1YN5C1-Q",
        "outputId": "2bd8830d-f357-4c9c-a4db-4339d7630830"
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      "cell_type": "code",
      "source": [
        "analyzer_instance.summary_metrics().roi.sel({\"distribution\": \"posterior\"}).drop_vars(\"distribution\").to_dataframe().unstack()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                   roi                              \n",
              "metric            mean    median     ci_lo     ci_hi\n",
              "channel                                             \n",
              "Channel0      1.586766  1.463530  0.431896  3.155697\n",
              "Channel1      0.936712  0.815354  0.261797  1.977477\n",
              "Channel2      1.418973  1.346149  0.543670  2.552366\n",
              "Channel3      0.876884  0.802209  0.297994  1.710027\n",
              "Channel4      1.620968  1.495501  0.405906  3.291715\n",
              "All Channels  1.208054  1.200087  0.800349  1.648818"
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      "source": [
        "<a name=\"roi-mroi-distributions\"></a>\n",
        "\n",
        "### ROI and mROI Posterior Distributions as Tensors\n",
        "\n",
        "A key benefit of Meridian's Bayesian approach is that it doesn't just give you one number for ROI. Instead, it gives you a full **posterior distribution** for each metric, which represents the range of likely values given your data.\n",
        "\n",
        "We can get these distributions for both ROI and mROI using the `.roi()` and `.marginal_roi()` methods."
      ]
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        "# Calculate ROI and mROI posterior distributions (as TensorFlow Tensors)\n",
        "mroi_dist = analyzer_instance.marginal_roi()\n",
        "roi_dist = analyzer_instance.roi()\n",
        "\n",
        "print(\"Shape of the ROI posterior distribution tensor:\", roi_dist.shape)\n",
        "print(\"Shape of the mROI posterior distribution tensor:\", mroi_dist.shape)\n",
        "print(\"\\nThe shape represents (Number of Chains, Number of Draws, Number of Geos, Number of Channels)\")"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Shape of the ROI posterior distribution tensor: (10, 500, 5)\n",
            "Shape of the mROI posterior distribution tensor: (10, 500, 5)\n",
            "\n",
            "The shape represents (Number of Chains, Number of Draws, Number of Geos, Number of Channels)\n"
          ]
        }
      ],
      "execution_count": null
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name=\"credible-intervals\"></a>\n",
        "### Standard Deviations of mROI & ROI Posterior Distributions"
      ],
      "metadata": {
        "id": "FJvLOrwFzjnt"
      }
    },
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      "cell_type": "markdown",
      "source": [
        "The full posterior distribution can be used to construct any summary metrics of interest. For example, the posterior standard deviation can be obtained as follows."
      ],
      "metadata": {
        "id": "vBh44MzTVUfR"
      }
    },
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        "id": "Ue6SxiJVTt9",
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      },
      "source": [
        "roi_stdev = np.std(roi_dist, (0, 1))\n",
        "mroi_stdev = np.std(mroi_dist, (0, 1))\n",
        "\n",
        "stdev_df = pd.DataFrame({\n",
        "    \"ROI Stdev\": roi_stdev,\n",
        "    \"MROI Stdev\": mroi_stdev\n",
        "    }, index=channels)\n",
        "print(\"Posterior Standard Deviations:\")\n",
        "print(stdev_df)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Posterior Standard Deviations:\n",
            "          ROI Stdev  MROI Stdev\n",
            "Channel0   0.860600    0.442567\n",
            "Channel1   0.557213    0.280965\n",
            "Channel2   0.620319    0.367566\n",
            "Channel3   0.438754    0.208942\n",
            "Channel4   0.903968    0.394786\n"
          ]
        }
      ]
    },
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      "cell_type": "markdown",
      "metadata": {
        "id": "9e085e8a"
      },
      "source": [
        "### Posterior Probability of ROI > 1\n",
        "\n",
        "Another useful metric is the posterior probability that a channel's ROI is greater than 1. This directly tells us the probability that a channel is profitable, given the data and model.\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "(mmm.inference_data.posterior.roi_m >= 1).mean(dim=('chain', 'draw'))"
      ],
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        "id": "w0gKy19noHjY",
        "outputId": "3008e45d-95f3-4ff9-d2d8-00a50cac5c21"
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      "execution_count": null,
      "outputs": [
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              "  padding-left: 0 !important;\n",
              "  display: grid;\n",
              "  grid-template-columns: 150px auto auto 1fr 0 20px 0 20px;\n",
              "}\n",
              "\n",
              ".xr-section-item {\n",
              "  display: contents;\n",
              "}\n",
              "\n",
              ".xr-section-item input {\n",
              "  display: inline-block;\n",
              "  opacity: 0;\n",
              "  height: 0;\n",
              "}\n",
              "\n",
              ".xr-section-item input + label {\n",
              "  color: var(--xr-disabled-color);\n",
              "  border: 2px solid transparent !important;\n",
              "}\n",
              "\n",
              ".xr-section-item input:enabled + label {\n",
              "  cursor: pointer;\n",
              "  color: var(--xr-font-color2);\n",
              "}\n",
              "\n",
              ".xr-section-item input:focus + label {\n",
              "  border: 2px solid var(--xr-font-color0) !important;\n",
              "}\n",
              "\n",
              ".xr-section-item input:enabled + label:hover {\n",
              "  color: var(--xr-font-color0);\n",
              "}\n",
              "\n",
              ".xr-section-summary {\n",
              "  grid-column: 1;\n",
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              "  font-weight: 500;\n",
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              "\n",
              ".xr-section-summary > span {\n",
              "  display: inline-block;\n",
              "  padding-left: 0.5em;\n",
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              "\n",
              ".xr-section-summary-in:disabled + label {\n",
              "  color: var(--xr-font-color2);\n",
              "}\n",
              "\n",
              ".xr-section-summary-in + label:before {\n",
              "  display: inline-block;\n",
              "  content: \"►\";\n",
              "  font-size: 11px;\n",
              "  width: 15px;\n",
              "  text-align: center;\n",
              "}\n",
              "\n",
              ".xr-section-summary-in:disabled + label:before {\n",
              "  color: var(--xr-disabled-color);\n",
              "}\n",
              "\n",
              ".xr-section-summary-in:checked + label:before {\n",
              "  content: \"▼\";\n",
              "}\n",
              "\n",
              ".xr-section-summary-in:checked + label > span {\n",
              "  display: none;\n",
              "}\n",
              "\n",
              ".xr-section-summary,\n",
              ".xr-section-inline-details {\n",
              "  padding-top: 4px;\n",
              "  padding-bottom: 4px;\n",
              "}\n",
              "\n",
              ".xr-section-inline-details {\n",
              "  grid-column: 2 / -1;\n",
              "}\n",
              "\n",
              ".xr-section-details {\n",
              "  display: none;\n",
              "  grid-column: 1 / -1;\n",
              "  margin-bottom: 5px;\n",
              "}\n",
              "\n",
              ".xr-section-summary-in:checked ~ .xr-section-details {\n",
              "  display: contents;\n",
              "}\n",
              "\n",
              ".xr-array-wrap {\n",
              "  grid-column: 1 / -1;\n",
              "  display: grid;\n",
              "  grid-template-columns: 20px auto;\n",
              "}\n",
              "\n",
              ".xr-array-wrap > label {\n",
              "  grid-column: 1;\n",
              "  vertical-align: top;\n",
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              "\n",
              ".xr-preview {\n",
              "  color: var(--xr-font-color3);\n",
              "}\n",
              "\n",
              ".xr-array-preview,\n",
              ".xr-array-data {\n",
              "  padding: 0 5px !important;\n",
              "  grid-column: 2;\n",
              "}\n",
              "\n",
              ".xr-array-data,\n",
              ".xr-array-in:checked ~ .xr-array-preview {\n",
              "  display: none;\n",
              "}\n",
              "\n",
              ".xr-array-in:checked ~ .xr-array-data,\n",
              ".xr-array-preview {\n",
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              "}\n",
              "\n",
              ".xr-dim-list {\n",
              "  display: inline-block !important;\n",
              "  list-style: none;\n",
              "  padding: 0 !important;\n",
              "  margin: 0;\n",
              "}\n",
              "\n",
              ".xr-dim-list li {\n",
              "  display: inline-block;\n",
              "  padding: 0;\n",
              "  margin: 0;\n",
              "}\n",
              "\n",
              ".xr-dim-list:before {\n",
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              "\n",
              ".xr-dim-list:after {\n",
              "  content: \")\";\n",
              "}\n",
              "\n",
              ".xr-dim-list li:not(:last-child):after {\n",
              "  content: \",\";\n",
              "  padding-right: 5px;\n",
              "}\n",
              "\n",
              ".xr-has-index {\n",
              "  font-weight: bold;\n",
              "}\n",
              "\n",
              ".xr-var-list,\n",
              ".xr-var-item {\n",
              "  display: contents;\n",
              "}\n",
              "\n",
              ".xr-var-item > div,\n",
              ".xr-var-item label,\n",
              ".xr-var-item > .xr-var-name span {\n",
              "  background-color: var(--xr-background-color-row-even);\n",
              "  border-color: var(--xr-background-color-row-odd);\n",
              "  margin-bottom: 0;\n",
              "  padding-top: 2px;\n",
              "}\n",
              "\n",
              ".xr-var-item > .xr-var-name:hover span {\n",
              "  padding-right: 5px;\n",
              "}\n",
              "\n",
              ".xr-var-list > li:nth-child(odd) > div,\n",
              ".xr-var-list > li:nth-child(odd) > label,\n",
              ".xr-var-list > li:nth-child(odd) > .xr-var-name span {\n",
              "  background-color: var(--xr-background-color-row-odd);\n",
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              "\n",
              ".xr-var-name {\n",
              "  grid-column: 1;\n",
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              "\n",
              ".xr-var-dims {\n",
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              "\n",
              ".xr-var-dtype {\n",
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              "\n",
              ".xr-var-preview {\n",
              "  grid-column: 4;\n",
              "}\n",
              "\n",
              ".xr-index-preview {\n",
              "  grid-column: 2 / 5;\n",
              "  color: var(--xr-font-color2);\n",
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              "\n",
              ".xr-var-name,\n",
              ".xr-var-dims,\n",
              ".xr-var-dtype,\n",
              ".xr-preview,\n",
              ".xr-attrs dt {\n",
              "  white-space: nowrap;\n",
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              "\n",
              ".xr-var-name:hover,\n",
              ".xr-var-dims:hover,\n",
              ".xr-var-dtype:hover,\n",
              ".xr-attrs dt:hover {\n",
              "  overflow: visible;\n",
              "  width: auto;\n",
              "  z-index: 1;\n",
              "}\n",
              "\n",
              ".xr-var-attrs,\n",
              ".xr-var-data,\n",
              ".xr-index-data {\n",
              "  display: none;\n",
              "  border-top: 2px dotted var(--xr-background-color);\n",
              "  padding-bottom: 20px !important;\n",
              "  padding-top: 10px !important;\n",
              "}\n",
              "\n",
              ".xr-var-attrs-in + label,\n",
              ".xr-var-data-in + label,\n",
              ".xr-index-data-in + label {\n",
              "  padding: 0 1px;\n",
              "}\n",
              "\n",
              ".xr-var-attrs-in:checked ~ .xr-var-attrs,\n",
              ".xr-var-data-in:checked ~ .xr-var-data,\n",
              ".xr-index-data-in:checked ~ .xr-index-data {\n",
              "  display: block;\n",
              "}\n",
              "\n",
              ".xr-var-data > table {\n",
              "  float: right;\n",
              "}\n",
              "\n",
              ".xr-var-data > pre,\n",
              ".xr-index-data > pre,\n",
              ".xr-var-data > table > tbody > tr {\n",
              "  background-color: transparent !important;\n",
              "}\n",
              "\n",
              ".xr-var-name span,\n",
              ".xr-var-data,\n",
              ".xr-index-name div,\n",
              ".xr-index-data,\n",
              ".xr-attrs {\n",
              "  padding-left: 25px !important;\n",
              "}\n",
              "\n",
              ".xr-attrs,\n",
              ".xr-var-attrs,\n",
              ".xr-var-data,\n",
              ".xr-index-data {\n",
              "  grid-column: 1 / -1;\n",
              "}\n",
              "\n",
              "dl.xr-attrs {\n",
              "  padding: 0;\n",
              "  margin: 0;\n",
              "  display: grid;\n",
              "  grid-template-columns: 125px auto;\n",
              "}\n",
              "\n",
              ".xr-attrs dt,\n",
              ".xr-attrs dd {\n",
              "  padding: 0;\n",
              "  margin: 0;\n",
              "  float: left;\n",
              "  padding-right: 10px;\n",
              "  width: auto;\n",
              "}\n",
              "\n",
              ".xr-attrs dt {\n",
              "  font-weight: normal;\n",
              "  grid-column: 1;\n",
              "}\n",
              "\n",
              ".xr-attrs dt:hover span {\n",
              "  display: inline-block;\n",
              "  background: var(--xr-background-color);\n",
              "  padding-right: 10px;\n",
              "}\n",
              "\n",
              ".xr-attrs dd {\n",
              "  grid-column: 2;\n",
              "  white-space: pre-wrap;\n",
              "  word-break: break-all;\n",
              "}\n",
              "\n",
              ".xr-icon-database,\n",
              ".xr-icon-file-text2,\n",
              ".xr-no-icon {\n",
              "  display: inline-block;\n",
              "  vertical-align: middle;\n",
              "  width: 1em;\n",
              "  height: 1.5em !important;\n",
              "  stroke-width: 0;\n",
              "  stroke: currentColor;\n",
              "  fill: currentColor;\n",
              "}\n",
              "\n",
              ".xr-var-attrs-in:checked + label > .xr-icon-file-text2,\n",
              ".xr-var-data-in:checked + label > .xr-icon-database,\n",
              ".xr-index-data-in:checked + label > .xr-icon-database {\n",
              "  color: var(--xr-font-color0);\n",
              "  filter: drop-shadow(1px 1px 5px var(--xr-font-color2));\n",
              "  stroke-width: 0.8px;\n",
              "}\n",
              "</style><pre class='xr-text-repr-fallback'>&lt;xarray.DataArray &#x27;roi_m&#x27; (media_channel: 5)&gt; Size: 40B\n",
              "array([0.72  , 0.3732, 0.735 , 0.3344, 0.7298])\n",
              "Coordinates:\n",
              "  * media_channel  (media_channel) object 40B &#x27;Channel0&#x27; ... &#x27;Channel4&#x27;</pre><div class='xr-wrap' style='display:none'><div class='xr-header'><div class='xr-obj-type'>xarray.DataArray</div><div class='xr-array-name'>'roi_m'</div><ul class='xr-dim-list'><li><span class='xr-has-index'>media_channel</span>: 5</li></ul></div><ul class='xr-sections'><li class='xr-section-item'><div class='xr-array-wrap'><input id='section-6a6b5065-0216-44a0-be97-eeb0cd12f5ee' class='xr-array-in' type='checkbox' checked><label for='section-6a6b5065-0216-44a0-be97-eeb0cd12f5ee' title='Show/hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-array-preview xr-preview'><span>0.72 0.3732 0.735 0.3344 0.7298</span></div><div class='xr-array-data'><pre>array([0.72  , 0.3732, 0.735 , 0.3344, 0.7298])</pre></div></div></li><li class='xr-section-item'><input id='section-e4c1fb47-0216-413a-8654-4c45a11ae9c3' class='xr-section-summary-in' type='checkbox'  checked><label for='section-e4c1fb47-0216-413a-8654-4c45a11ae9c3' class='xr-section-summary' >Coordinates: <span>(1)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'><li class='xr-var-item'><div class='xr-var-name'><span class='xr-has-index'>media_channel</span></div><div class='xr-var-dims'>(media_channel)</div><div class='xr-var-dtype'>object</div><div class='xr-var-preview xr-preview'>&#x27;Channel0&#x27; ... &#x27;Channel4&#x27;</div><input id='attrs-255619bb-2633-4561-b6be-7de555ad0c65' class='xr-var-attrs-in' type='checkbox' disabled><label for='attrs-255619bb-2633-4561-b6be-7de555ad0c65' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-a67e9f67-e490-469f-8c62-3a4c60cdcc89' class='xr-var-data-in' type='checkbox'><label for='data-a67e9f67-e490-469f-8c62-3a4c60cdcc89' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'></dl></div><div class='xr-var-data'><pre>array([&#x27;Channel0&#x27;, &#x27;Channel1&#x27;, &#x27;Channel2&#x27;, &#x27;Channel3&#x27;, &#x27;Channel4&#x27;],\n",
              "      dtype=object)</pre></div></li></ul></div></li><li class='xr-section-item'><input id='section-500e9622-0084-429e-a237-2c6ce057d35c' class='xr-section-summary-in' type='checkbox'  ><label for='section-500e9622-0084-429e-a237-2c6ce057d35c' class='xr-section-summary' >Indexes: <span>(1)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'><li class='xr-var-item'><div class='xr-index-name'><div>media_channel</div></div><div class='xr-index-preview'>PandasIndex</div><input type='checkbox' disabled/><label></label><input id='index-edbd4ad4-b301-497d-86cc-9fddb2effaed' class='xr-index-data-in' type='checkbox'/><label for='index-edbd4ad4-b301-497d-86cc-9fddb2effaed' title='Show/Hide index repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-index-data'><pre>PandasIndex(Index([&#x27;Channel0&#x27;, &#x27;Channel1&#x27;, &#x27;Channel2&#x27;, &#x27;Channel3&#x27;, &#x27;Channel4&#x27;], dtype=&#x27;object&#x27;, name=&#x27;media_channel&#x27;))</pre></div></li></ul></div></li><li class='xr-section-item'><input id='section-66d55c9a-d14d-4a6c-86c0-3eb9ebede497' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-66d55c9a-d14d-4a6c-86c0-3eb9ebede497' class='xr-section-summary'  title='Expand/collapse section'>Attributes: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><dl class='xr-attrs'></dl></div></li></ul></div></div>"
            ]
          },
          "metadata": {},
          "execution_count": 22
        }
      ]
    },
    {
      "metadata": {
        "id": "DgTfqgDpPiYK"
      },
      "cell_type": "markdown",
      "source": [
        "<a name=\"response-curves\"></a>\n",
        "### Visualizing Performance with Response Curves\n",
        "\n",
        "**Response curves** help you visualize how each channel saturates as a function of spend. The curve shows how the incremental outcome (revenue, in this example) for a channel increases as you spend more. A flattening curve indicates diminishing returns.\n",
        "\n",
        "We can plot these curves easily using the `visualizer` module."
      ]
    },
    {
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 882
        },
        "id": "cGbPHv7-C6bK",
        "outputId": "5d76f9d9-df65-44e6-a40e-4695f50cc4ee"
      },
      "cell_type": "code",
      "source": [
        "# Create a MediaEffects visualizer object\n",
        "media_effects = visualizer.MediaEffects(mmm)\n",
        "\n",
        "# Plot the response curves for each channel\n",
        "media_effects.plot_response_curves(\n",
        "    plot_separately=True,\n",
        "    include_ci=True # Shows the credible interval on the curve\n",
        ")"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
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